# How do you find the exact values of cos(5pi/12) using the half angle formula?

##### 2 Answers

#### Explanation:

By the half angle formula:

If

Note that

so

Therefore

Since

#### Explanation:

How? First I sigh and point out that this is yet another trig problem that assumes the student knows only the two cliche triangles, 30/60/90 and 45/45/90.

You, dear reader, might object that this question only requires us to know one triangle, 30/60/90, which includes

There are a couple of things less than ideal here. First we have the ambiguous

Where does the minus sign come from? If instead of

The second issue is the nested square root. There's a theorem that says

We were asked to use the half angle formula. It would be better if we had used the sum angle formula. This gives an unambiguous result with no nested square root:

There's a bit of a mystery about how Ramanujan derived some of his radical expressions. Perhaps we get a hint in our first Ramanujan-like result: